Currently,
R and
S must both be polynomial rings over the same base field.
This function first checks to see whether M will be a finitely generated R-module via F. If not, an error message describing the codimension of M/(vars of S)M is given (this is equal to the dimension of R if and only if M is a finitely generated R-module.
Assuming that it is, the push forward
F_*(M) is computed. This is done by first finding a presentation for
M in terms of a set of elements that generates
M as an
S-module, and then applying the routine
coimage to a map whose target is
M and whose source is a free module over
R.
Example: The Auslander-Buchsbaum formula
Let's illustrate the Auslander-Buchsbaum formula. First construct some rings and make a module of projective dimension 2.
i1 : R4 = ZZ/32003[a..d];
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i2 : R5 = ZZ/32003[a..e];
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i3 : R6 = ZZ/32003[a..f];
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i4 : M = coker genericMatrix(R6,a,2,3)
o4 = cokernel | a c e |
| b d f |
2
o4 : R6-module, quotient of R6
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i5 : pdim M
o5 = 2
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Create ring maps.
i6 : G = map(R6,R5,{a+b+c+d+e+f,b,c,d,e})
o6 = map(R6,R5,{a + b + c + d + e + f, b, c, d, e})
o6 : RingMap R6 <--- R5
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i7 : F = map(R5,R4,random(R5^1, R5^{4:-1}))
o7 = map(R5,R4,{- 10855a - 13051b - 89c - 5042d + 9804e, - 8306a - 9626b - 180c + 9001d - 12123e, - 6871a + 7820b - 4571c - 5629d + 7068e, 9209a + 13479b + 13683c + 10675d + 13049e})
o7 : RingMap R5 <--- R4
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The module M, when thought of as an R5 or R4 module, has the same depth, but since depth M + pdim M = dim ring, the projective dimension will drop to 1, respectively 0, for these two rings.
i8 : P = pushForward(G,M)
o8 = cokernel | c -de |
| d bc-ad+bd+cd+d2+de |
2
o8 : R5-module, quotient of R5
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i9 : pdim P
o9 = 1
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i10 : Q = pushForward(F,P)
3
o10 = R4
o10 : R4-module, free, degrees {0, 1, 0}
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i11 : pdim Q
o11 = 0
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Example: generic projection of a homogeneous coordinate ring
We compute the pushforward N of the homogeneous coordinate ring M of the twisted cubic curve in P^3.
i12 : P3 = QQ[a..d];
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i13 : M = comodule monomialCurveIdeal(P3,{1,2,3})
o13 = cokernel | c2-bd bc-ad b2-ac |
1
o13 : P3-module, quotient of P3
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The result is a module with the same codimension, degree and genus as the twisted cubic, but the support is a cubic in the plane, necessarily having one node.
i14 : P2 = QQ[a,b,c];
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i15 : F = map(P3,P2,random(P3^1, P3^{-1,-1,-1}))
1 5 1 2 3 5 2 1
o15 = map(P3,P2,{-a + -b + 2c + -d, -a + -b + -c + d, -a + 2b + 3c + -d})
5 2 2 9 2 6 5 2
o15 : RingMap P3 <--- P2
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i16 : N = pushForward(F,M)
o16 = cokernel {0} | 55319112ab-181324008b2-33995260ac+186395184bc-45957470c2 3073284a2-25588548b2-5122814ac+36337050bc-10496464c2 1518327133398443664b3-2518113831084558000b2c-1265422363912700ac2+1396926714742443900bc2-258586217695231150c3 0 |
{1} | 239725928a+94543578b-267902999c 51748273a+10634517b-51904264c -2721429399335140674a2+580734999228300372ab-1344923712060566418b2+4767373365690659074ac+1270676213572985064bc-2673153309853354231c2 4610114a3-16094718a2b+16648902ab2-3072006b3-1828521a2c+11349648abc-6902091b2c-3317277ac2-3527532bc2+2669692c3 |
2
o16 : P2-module, quotient of P2
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i17 : hilbertPolynomial M
o17 = - 2*P + 3*P
0 1
o17 : ProjectiveHilbertPolynomial
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i18 : hilbertPolynomial N
o18 = - 2*P + 3*P
0 1
o18 : ProjectiveHilbertPolynomial
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i19 : ann N
3 2 2 3 2
o19 = ideal(4610114a - 16094718a b + 16648902a*b - 3072006b - 1828521a c +
-----------------------------------------------------------------------
2 2 2 3
11349648a*b*c - 6902091b c - 3317277a*c - 3527532b*c + 2669692c )
o19 : Ideal of P2
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Note: these examples are from the original Macaulay script by David Eisenbud.